Fundamental theorems of calculus and Zinbiel algebras

Derivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with a derivation and integration between them which together satisfy analogues of the two Fundamental Theorems of Calculus. In the special case of when an algebra is seen as a module over itself, we show that this sort of FTC-pair is precisely the same thing as an integro-differential algebra. We provide various constructions of FTC-pairs, as well as numerous examples of FTC-pairs including some based on polynomials, smooth functions, Hurwitz series, and shuffle algebras. Moreover, it is well known that integrations are closely related to Zinbiel algebras. The main result of this paper is that the category of FTC-pairs is equivalent to the category of Zinbiel algebras.